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Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$ The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\rho_j)_j,$ for a distribution $\Delta_jf:=\mathscr{F}^{-1}\rho_j*f:=K_j*f$): enter image description here Applying this theorem we get modification for stochastic convolution (for dimensions $d=1$) $X_r(\phi):=\int_0^r\int_{\mathbb{T}^1}(\phi,p(r-q,y-\cdot))\xi(dq,dy).$

By renormalizing and regularizing $|\partial_x X|^2$ we can prove that the limit is actually $U_r(\phi):=\int_{([0,r] \times \mathbb{T}^1)^2}(\phi,\partial_xp(r-q_1,y_1-\cdot)\partial_x p(r-q_2,y_2-\cdot))d\xi(q_1,y_1)d\xi(q_2,y_2), $(using Hermite polynomial $H_2$, we can express it in term of second Wiener-Ito integral, using Nelson inequality we deduce that the assumptions of the above theorem are verified).

Let $Y_r:=\int_0^rP_{r-q}(U_q)dq$ (Modifications of $Y$ in Besov spaces already exists since we already did the construction for the term inside).

By renormalizing and regularizing $W(r):= \partial_x X_r \partial_xY_r,$ what would be the limit expressed in term of iterated integrals?

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  • $\begingroup$ When $d \ge 2$, the Wick square of $\nabla X$ is not defined. When $d=1$, the construction of what you call $W$ is explained in my KPZ paper, but by now there are simpler and more general constructions. The "easiest" general construction is probably the one in my paper with Rhys Steele. $\endgroup$
    – Martin Hairer
    Commented Jun 26 at 6:41
  • $\begingroup$ @MartinHairer is it possible to indicate the reference for the paper (there are several one in collaboration with steele) $\endgroup$
    – mathex
    Commented Jun 26 at 14:27
  • $\begingroup$ The assumptions of that theorem are only satisfied in dimension 1 for the KPZ equation. $\endgroup$
    – Martin Hairer
    Commented Jun 27 at 5:06
  • $\begingroup$ Your image is presumably excerpted from somewhere. Where? $\endgroup$
    – LSpice
    Commented Jun 27 at 16:06

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